Random Graph Model for Structural Analysis of Online Communications

Ivan Sukharev and Maria Ivanova

International Conference on Software Testing, Machine Learning and Complex Process Analysis (TMPA-2019)
7-9 November 2019, Tbilisi

Video: https://youtu.be/ACbKae2kugI

TMPA Conference website https://tmpaconf.org/
TMPA Conference on Facebook https://www.facebook.com/groups/tmpaconf/

Transcript

1. Random Graph Model for Structural Analysis
   of Online Communications
   TMPA-2019
   Maria Ivanova
   Ivan Sukharev
   National Research University
   Higher School of Economics (Moscow)
   November 8, 2019
   

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2. Overview
   1. State of a problem
   2. Graph Definition
   3. Related works
   4. Adaptation of the Barabasi–Albert Growth Model
   5. New Random Graph Model
   6. Model Fitting
   Maria Ivanova Higher School of Economics November 8, 2019 2 / 14
   

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3. State of a problem
   Some problems:
   1. Processing costs
   2. Disclosure of personal
   information
   3. Statistical reliability
   Maria Ivanova Higher School of Economics November 8, 2019 3 / 14
   

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4. Graph Definition
   Let Vn be a set of vertices:
   Vn = {1, ..., n}
   Then a set of all edges En for
   Vn is as follows:
   En = {{i, j} | i, j ∈ Vn, i = j}
   A graph is an ordered pair
   G := (Vn, E)
   where E ⊂ En.
   Maria Ivanova Higher School of Economics November 8, 2019 4 / 14
   

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5. Related works
   Erdos-–Renyi Model
   The graph generation process consists in constructing a set of
   edges E for a given set of vertices Vn. The edge eij ∈ En is in
   the set of edges E of a random graph with probability p ∈ [0, 1].
   Maria Ivanova Higher School of Economics November 8, 2019 5 / 14
   

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6. Related works
   Scale-free network
   A scale-free network is a graph where the degree distribution of
   the vertices is described by a power-law, at least asymptotically.
   Therefore, the probability of a vertex having k edges at large
   values of k is proportional to k−γ:
   P (k) ∼ k−γ (1)
   Maria Ivanova Higher School of Economics November 8, 2019 6 / 14
   

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7. Related works
   Barabasi–Albert Growth Model
   A new vertex vn+1
   is added. Then, with probability pi
   , there is
   an edge between the new and the i-th vertices, where pi
   is
   calculated by the following formula:
   pi =
   deg (vi)
   n
   j=1
   deg vj
   (2)
   Maria Ivanova Higher School of Economics November 8, 2019 7 / 14
   

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8. Data
   We obtained 56003 articles
   with comments and
   constructed the comment
   graph for each of them.
   24% — the number of first
   level comments.
   Ci
   is a comment of ith level
   Maria Ivanova Higher School of Economics November 8, 2019 8 / 14
   

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9. Adaptation
   of the Barabasi–Albert Growth Model
   The probability of choosing the node is directly proportional to
   the number of edges attached to it. We add some parameter k
   to a root node degree, thereby increasing the likelihood of
   joining it rather than a comment.
   Maria Ivanova Higher School of Economics November 8, 2019 9 / 14
   

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10. New Random Graph Model
    Growth algorithm:
    1. With probability p, a new vertex joins the root of the tree,
    that is, the article itself. Its weight is recorded by the
    function φ, that is an indicator of interest to this message
    among other users.
    2. With probability of 1 − p, a new vertex joins any vertex at
    random, except for the root of the tree. The probability of
    joining each of them is proportional to their weights. A new
    vertex takes up λ from the weight of the vertex to which it is
    attached.
    Maria Ivanova Higher School of Economics November 8, 2019 10 / 14
    

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11. Model fitting
    Finding of parameter p
    The value of the first
    parameter p was calculated:
    24% of all the vertices are
    neighbors to the article.
    Maria Ivanova Higher School of Economics November 8, 2019 11 / 14
    

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12. Model fitting
    Parameter λ explanation
    When λ = 1, an entire weight
    of the vertex will go to the next
    level in the case of joining an
    edge. This indicates
    appearance of long leaves
    without branching.
    When λ = 0, a leaf will not go
    down beyond the second level
    and the nodes degrees in the
    first level will quickly grow.
    Maria Ivanova Higher School of Economics November 8, 2019 12 / 14
    

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13. Model fitting
    Finding of parameter λ
    The vertex that first joins the
    first-level comment takes up λ
    of its weight.
    The value of this parameter is
    λ = 0.7629.
    Since the subtree’s weight is
    proportional to the number of
    comments at the end of the
    discussion, the distribution of
    the random variable φ could
    be found from the subtree
    comments number.
    Maria Ivanova Higher School of Economics November 8, 2019 13 / 14
    

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14. Thank you for your attention!
    Random Graph Model for Structural
    Analysis of Online Communications
    Maria Ivanova
    Ivan Sukharev
    National Research University
    Higher School of Economics (Moscow)
    November 8, 2019
    Maria Ivanova Higher School of Economics November 8, 2019 14 / 14
    

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