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The paradox of second-order homophily in networks.

ABSTRACT: Homophily-the tendency of nodes to connect to others of the same type-is a central issue in the study of networks. Here we take a local view of homophily, defining notions of first-order homophily of a node (its individual tendency to link to similar others) and second-order homophily of a node (the aggregate first-order homophily of its neighbors). Through this view, we find a surprising result for homophily values that applies with only minimal assumptions on the graph topology. It can be phrased most simply as "in a graph of red and blue nodes, red friends of red nodes are on average more homophilous than red friends of blue nodes". This gap in averages defies simple intuitive explanations, applies to globally heterophilous and homophilous networks and is reminiscent of but structually distinct from the Friendship Paradox. The existence of this gap suggests intrinsic biases in homophily measurements between groups, and hence is relevant to empirical studies of homophily in networks.

SUBMITTER: Evtushenko A

PROVIDER: S-EPMC8233377 | biostudies-literature |

REPOSITORIES: biostudies-literature

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Project description:We present a new method for assessing and measuring homophily in networks whose nodes have categorical attributes, namely when the nodes of networks come partitioned into classes (colors). We probe this method in two different classes of networks: (i) protein-protein interaction (PPI) networks, where nodes correspond to proteins, partitioned according to their functional role, and edges represent functional interactions between proteins (ii) Pokec on-line social network, where nodes correspond to users, partitioned according to their age, and edges respresent friendship between users.Similarly to other classical and well consolidated approaches, our method compares the relative edge density of the subgraphs induced by each class with the corresponding expected relative edge density under a null model. The novelty of our approach consists in prescribing an endogenous null model, namely, the sample space of the null model is built on the input network itself. This allows us to give exact explicit expression for the [Formula: see text]-score of the relative edge density of each class as well as other related statistics. The [Formula: see text]-scores directly quantify the statistical significance of the observed homophily via Čebyšëv inequality. The expression of each [Formula: see text]-score is entered by the network structure through basic combinatorial invariant such as the number of subgraphs with two spanning edges. Each [Formula: see text]-score is computed in [Formula: see text] time for a network with n nodes and m edges. This leads to an overall efficient computational method for assesing homophily. We complement the analysis of homophily/heterophily by considering [Formula: see text]-scores of the number of isolated nodes in the subgraphs induced by each class, that are computed in O(nm) time. Theoretical results are then exploited to show that, as expected, both the analyzed network classes are significantly homophilic with respect to the considered node properties.

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The paradox of second-order homophily in networks.

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