The soft consensus model: dynamics, complexity, and modularity G. Campanella1 T. Ara´ ujo2 R. A. Marques Pereira3 1Faculty of Science and Technology (FCT), New University of Lisbon 2ISEG (School of Economics and Management), Technical University of Lisbon and Research Unit on Complexity in Economics (UECE) 3Department of Computer and Management Sciences, University of Trento ECCS’10 European Conference on Complex Systems September 17, 2010 — Lisbon, Portugal 

Contents 1 The soft consensus model 2 Complexity and modularity 3 Results 

Contents 1 The soft consensus model 2 Complexity and modularity 3 Results 

The soft consensus model Network model • Dynamic opinions ri ∈ [0, 1] are encoded by primary vertices; • Static opinions si ∈ [0, 1] are encoded by secondary vertices; • The consensual interaction between primary vertices is modulated by the interaction coefficient vij ∈ (0, 1); • The inertial interaction between primary and associated secondary vertices is modulated by the interaction coefficient ui ∈ (0, 1). Larger interaction values correspond to stronger interactions. 

The soft consensus model Example with ten agents r1 s1 r2 s2 r3 s3 r4 s4 r5 s5 r6 s6 r7 s7 r8 s8 r9 s9 r10 s10 

The soft consensus model Consensual network dynamics The dynamical process of opinion change is driven by minimization of a cost function W = (1 − λ) V + λU, λ ∈ [0, 1], combining: • a measure V of the overall dissensus in the present network configuration; • a measure U of the overall change from the original network configuration (leading to opinion changing aversion). It acts on the individual opinion variables ri through the iterative process ri ri = ri − γ ∂W ∂ri and can thus be regarded as an unsupervised learning algorithm. 

The soft consensus model Consensual network dynamics The dynamical process of opinion change is driven by minimization of a cost function W = (1 − λ) V + λU, λ ∈ [0, 1], combining: • a measure V of the overall dissensus in the present network configuration; • a measure U of the overall change from the original network configuration (leading to opinion changing aversion). It acts on the individual opinion variables ri through the iterative process ri ri = ri − γ ∂W ∂ri and can thus be regarded as an unsupervised learning algorithm. The objective is maximum consensus with minimum change. 

The soft consensus model Non-linearity of measures Both V and U are non-linear measures, V = 1 4 i Vi, Vi = j=i f (ri − rj)2 n − 1 U = 1 2 i Ui, Ui = f (ri − si)2 where f is the scaling function f(x) = − 1 β ln 1 + e−β(x−α) , 0 < α < 1, β > 0. 

The soft consensus model The scaling function f -0.3 -0.2 -0.1 0 0.1 -0.1 0 0.1 0.2 0.3 0.4 β = 10 β = 20 β = 50 Graph of f(x) = − 1 β ln 1 + e−β(x−0.04) 

The soft consensus model Interaction coefficients The new opinions r i are obtained as a convex combination of • the current opinion ri; • the mean opinion of the rest of the group ¯ ri = j=i vijrj j=i vij ; • the original opinion si. Its coefficients are based on vi = j=i vij n−1 and ui, which are recomputed at each iteration as vij = f (ri − rj)2 and ui = f (ri − si)2 , where f is the derivative of the scaling function f, f (x) = 1 1 + eβ(x−α) , 0 < α < 1, β > 0. 

The soft consensus model The sigmoid function f f is a decreasing sigmoid function with parameters • α representing the square of the critical interaction distance; • β controlling the polarization of the sigmoid function. -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 β = 10 β = 20 β = 50 

The soft consensus model Simulation with ten agents 

The soft consensus model Summary 1. Each agent is associated with a primary vertex, encoding its dynamical opinion, and a secondary vertex, encoding its static, original opinion; 2. The strength of both consensual and inertial interactions depends (non-linearly) on the Euclidean distance between them. 3. The dynamical process is driven by minimization of a cost function combining overall dissensus and change, and thus aims at achieving maximum consensus with minimum change. 

Contents 1 The soft consensus model 2 Complexity and modularity 3 Results 

Partition • It is an equivalence relation over the set of vertices; • It can be represented as another undirected, unweighted graph with the same vertices and edges xij ∈ {0, 1}. To ensure consistency, the following constraints must be enforced for all triples of vertices, with i, j, k = 1, . . . , n, xii = 1 (reflexivity) xij = xji (symmetry)      xij + xjk − 2xik ≤ 1 xik + xij − 2xjk ≤ 1 xjk + xik − 2xij ≤ 1 (transitivity) 

Partition • It is an equivalence relation over the set of vertices; • It can be represented as another undirected, unweighted graph with the same vertices and edges xij ∈ {0, 1}. To ensure consistency, the following constraints must be enforced for all triples of vertices, with i, j, k = 1, . . . , n, xii = 1 (reflexivity) xij = xji (symmetry)      xij + xjk − 2xik ≤ 1 xik + xij − 2xjk ≤ 1 xjk + xik − 2xij ≤ 1 (transitivity) Complexity and modularity are both functions of a particular partition of a network. 

Complexity • Given a weighted graph, we can turn it into a partition by means of two operations defined on edges: Completion if the element xij is set to 1; Deletion if the element xij is set to 0. • Given a weighted graph and a specific partition, complexity is a scalar value measuring how “close” the partition is to the original graph. 

Complexity • Given a weighted graph, we can turn it into a partition by means of two operations defined on edges: Completion if the element xij is set to 1; Deletion if the element xij is set to 0. • Given a weighted graph and a specific partition, complexity is a scalar value measuring how “close” the partition is to the original graph. C = i,j (1 − aij) xij + aij (1 − xij) 

Complexity • Given a weighted graph, we can turn it into a partition by means of two operations defined on edges: Completion if the element xij is set to 1; Deletion if the element xij is set to 0. • Given a weighted graph and a specific partition, complexity is a scalar value measuring how “close” the partition is to the original graph. C = i,j (1 − aij) xij + aij (1 − xij) 1. The cost of completing edge aij is (1 − aij); 

Complexity • Given a weighted graph, we can turn it into a partition by means of two operations defined on edges: Completion if the element xij is set to 1; Deletion if the element xij is set to 0. • Given a weighted graph and a specific partition, complexity is a scalar value measuring how “close” the partition is to the original graph. C = i,j (1 − aij) xij + aij (1 − xij) 1. The cost of completing edge aij is (1 − aij); 2. The cost of deleting edge aij is aij. 

Modularity • Given a weighted graph and a specific partition, modularity is a scalar value measuring how much “above randomness” each intra-group edge is. • It does so by comparing the number of edges within equivalence classes and the number of such edges one would expect in a random graph of the same order and with the same degree distribution (null model). 

Modularity • Given a weighted graph and a specific partition, modularity is a scalar value measuring how much “above randomness” each intra-group edge is. • It does so by comparing the number of edges within equivalence classes and the number of such edges one would expect in a random graph of the same order and with the same degree distribution (null model). Q = 1 a i,j aij − ai aj a xij, ai = j aij , a = i ai 

Modularity • Given a weighted graph and a specific partition, modularity is a scalar value measuring how much “above randomness” each intra-group edge is. • It does so by comparing the number of edges within equivalence classes and the number of such edges one would expect in a random graph of the same order and with the same degree distribution (null model). Q = 1 a i,j aij − ai aj a xij, ai = j aij , a = i ai For all edges in the same equivalence class, 

Modularity • Given a weighted graph and a specific partition, modularity is a scalar value measuring how much “above randomness” each intra-group edge is. • It does so by comparing the number of edges within equivalence classes and the number of such edges one would expect in a random graph of the same order and with the same degree distribution (null model). Q = 1 a i,j aij − ai aj a xij, ai = j aij , a = i ai For all edges in the same equivalence class, consider how much the actual edge value is above the one predicted by the null model. 

Modularity • Given a weighted graph and a specific partition, modularity is a scalar value measuring how much “above randomness” each intra-group edge is. • It does so by comparing the number of edges within equivalence classes and the number of such edges one would expect in a random graph of the same order and with the same degree distribution (null model). Q = 1 a i,j aij − ai aj a xij, ai = j aij , a = i ai For all edges in the same equivalence class, consider how much the actual edge value is above the one predicted by the null model. The null model implicitly assumes interaction among all pairs of vertices, even loops! 

Complexity and modularity Extreme values for scaling To allow for a better comparison, the two measures have been linearly scaled to the unit interval. Modularity The minimal and maximal values attainable in unweighted graphs are −1 2 and 1, respectively. Complexity The minimal value is certainly 0. As for the maximum, if we do not take into account the transitivity constraints, we have that C ≤ i,j max(aij , 1 − aij). In our simulations, we used the highest complexity obtained by analyzing all bipartite graphs of order 8, since they achieve a minimal modularity value. 

Complexity and modularity Optimization using ILP Complexity and modularity are both functions of a particular partition of a network. 

Complexity and modularity Optimization using ILP Complexity and modularity are both functions of a particular partition of a network. The problem of finding the optimal partition can be tackled by means of Integer Linear Programming (ILP) but this approach severely limits the size of instances that can be taken into consideration. 

Complexity and modularity Summary 1. Partitions are equivalence relations over the set of vertices that can also be be represented as graphs; 2. Complexity measures the distance of a partition to the original graph, while modularity measures how much “above randomness” intra-group edges are by predicting and subtracting the “background noise” expected within each group, as given by the null model; 3. The optimal partitions for both measures can be found using Integer Linear Programming, though this limits the size of tractable instances. 

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Contents 1 The soft consensus model 2 Complexity and modularity 3 Results 

Case study Scenario • 8 vertices divided in k small groups of variable size, each centered at cj = j k+1 , j = 1, . . . , k. • Initial ri values are drawn from a Normal distribution with mean cj and standard deviation 4 k+1 . • Simulation stops when all new ri values differ less than 10−6 from previous ones. • The sigmoid function parameters are: • α = 0.04, yielding a critical interaction distance of 0.2; • β ≈ 114.88, determined such that f (0) = 0.99. 

Case study Scenario • 8 vertices divided in k small groups of variable size, each centered at cj = j k+1 , j = 1, . . . , k. • Initial ri values are drawn from a Normal distribution with mean cj and standard deviation 4 k+1 . • Simulation stops when all new ri values differ less than 10−6 from previous ones. • The sigmoid function parameters are: • α = 0.04, yielding a critical interaction distance of 0.2; • β ≈ 114.88, determined such that f (0) = 0.99. We will only analyze the 4+1+1+2 case! 

Case study Evolution of preferences 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1000 2000 3000 4000 5000 6000 Preference Iteration 

Case study Evolution of measures 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1000 2000 3000 4000 5000 6000 Measure Iteration Modularity Complexity 

Summary 1. The soft consensus model is a network-based dynamic system driven by a simple force that exhibits complex behavior. 2. Complexity is an intuitive measure of distance between a partition and the original network, while modularity is a measure of how much intra-group edges are “above randomness” (finding exact optimal partitions for both measures can be done using ILP). 3. Complexity and modularity are both able to capture interaction phenomena occurring in the soft consensus model. 

Food for thought 1. How are complexity and modularity related? 2. What other systems could be studied using a similar approach? 3. What other measures could help us better understand the complexity of the soft consensus model? 

Thank you for your attention! Questions? 

Essential bibliography How it all started... T. Ara´ ujo and R. A. Marques Pereira. A measure of complexity in the soft consensus model of group decision making. In Proceedings of the 8th International Conference on Information Processing and Management of Uncertainty, volume 1, pages 621–625, Madrid, Spain, July 2000. Technical University of Madrid. 

Essential bibliography I Soft consensus model J. Kacprzyk and M. Fedrizzi. A ‘soft’ measure of consensus in the setting of partial (fuzzy) preferences. European Journal of Operational Research, 34(3):316–325, 1988. J. Kacprzyk and M. Fedrizzi. A ‘human-consistent’ degree of consensus based on fuzzy logic with linguistic quantifiers. Mathematical Social Sciences, 18(3):275–290, 1989. J. Kacprzyk, M. Fedrizzi, and H. Nurmi. Group decision making and consensus under fuzzy preferences and fuzzy majority. Fuzzy Sets and Systems, 49(1):21–31, 1992. 

Essential bibliography II Soft consensus model M. Fedrizzi, M. Fedrizzi, R. A. Marques Pereira, and A. Zorat. A dynamical model for reaching consensus in group decision making. In SAC ’95: Proceedings of the 1995 ACM Symposium on Applied Computing, pages 493–496, New York, NY, USA, 1995. ACM. M. Fedrizzi, M. Fedrizzi, R. A. Marques Pereira, and C. Ribaga. Consensual dynamics with local externalities. In Proceedings of the 1st International Workshop on Preferences and Decisions TRENTO’97, pages 62–69, Trento, Italy, June 1997. 

Essential bibliography III Soft consensus model M. Fedrizzi, M. Fedrizzi, and R. A. Marques Pereira. Soft consensus and network dynamics in group decision making. International Journal of Intelligent Systems, 14(1):63–77, 1999. M. Fedrizzi, M. Fedrizzi, and R. A. Marques Pereira. Consensus modelling in group decision making: a dynamical approach based on fuzzy preferences. New Mathematics and Natural Computation, 3(2):219–237, 2007. 

Essential bibliography I Modularity M. E. J. Newman. Detecting community structure in networks. The European Physical Journal B, 38(2):321–330, 2004. M. E. J. Newman. Fast algorithm for detecting community structure in networks. Physical Review E, 69(6):66133–66138, 2004. M. E. J. Newman and M. Girvan. Finding and evaluating community structure in networks. Physical Review E, 69(2):26113–26128, 2004. M. E. J. Newman. Finding community structure in networks using the eigenvectors of matrices. Physical Review E, 74(3):36104–36123, 2006. 

Essential bibliography II Modularity M. E. J. Newman. Modularity and community structure in networks. Proceedings of the National Academy of Sciences, 103(23):8577–8582, 2006. S. Fortunato and M. Barth´ elemy. Resolution limit in community detection. Proceedings of the National Academy of Sciences, 104(1):36–41, 2007. U. Brandes, D. Delling, M. Gaertler, R. G¨ orke, M. Hoefer, Z. Nikoloski, and D. Wagner. On modularity clustering. IEEE Transactions on Knowledge and Data Engineering, 20(2):172–188, 2008.