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
∈ [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.
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.
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.
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.
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.
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.
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)
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.
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.
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)
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);
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 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 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 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 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 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!
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.
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.
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.
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.
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!
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.
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?
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