Tsallis entropy of complex networks

How complex of the complex networks has attracted many researchers to explore it. The entropy is an useful method to describe the degree of the $complex$ of the complex networks. In this paper, a new method which is based on the Tsallis entropy is proposed to describe the $complex$ of the complex networks. The results in this paper show that the complex of the complex networks not only decided by the structure property of the complex networks, but also influenced by the relationship between each nodes. In other word, which kinds of nodes are chosen as the main part of the complex networks will influence the value of the entropy of the complex networks. The value of q in the Tsallis entropy of the complex networks is used to decided which kinds of nodes will be chosen as the main part in the complex networks. The proposed Tsallis entropy of the complex networks is a generalised method to describe the property of the complex networks.

💡 Research Summary

The paper introduces a novel method for quantifying the complexity of complex networks by employing Tsallis entropy, a generalization of the traditional Shannon entropy that incorporates a tunable non‑additivity parameter q. The authors begin by reviewing the importance of complex networks across disciplines and the limitations of existing entropy‑based measures, which typically rely solely on degree distributions and treat all nodes uniformly. To overcome this, they first define a normalized degree d_i for each node (the node’s degree divided by the sum of all degrees) and treat these normalized degrees as probabilities p_i.

Tsallis entropy is then introduced in its standard form S_q = k (1 − ∑p_i^q)/(q − 1), where q controls the degree of non‑extensivity; when q = 1, the expression reduces to the Shannon entropy. Building on this, the authors propose a network‑specific Tsallis entropy:

 S_q′ = k (1 − ∑d_i^q / n)/(q − 1),

where n is the total number of nodes. This formulation directly links the entropy to the degree heterogeneity of the network while allowing the parameter q to bias the contribution of nodes with different degrees.

The paper systematically analyzes the behavior of S_q′ for several regimes of q:

* q = 0: every node contributes equally, yielding the maximal entropy and representing the most “complex” configuration.
* 0 < q < 1: low‑degree nodes receive higher weight, increasing the entropy and emphasizing the role of peripheral structures.
* q = 1: the measure collapses to the conventional degree‑based Shannon entropy, reflecting purely structural complexity.
* q > 1: high‑degree (hub) nodes dominate, driving the entropy toward zero and indicating an increasingly ordered network.
* q → ∞: only the highest‑degree nodes matter, and the entropy asymptotically approaches zero.

To demonstrate practical relevance, the authors apply the metric to four real‑world networks: the US‑Airlines route network, an email communication network, the German highway system, and a yeast protein‑protein interaction (PPI) network. For each network, they compute S_q′ across a range of q values (0.5 to 2.1). The results reveal consistent patterns: larger networks exhibit a more pronounced dependence of entropy on q; networks dominated by many low‑degree nodes show a steep increase in entropy as q decreases below 1, whereas hub‑centric networks display a rapid decline in entropy as q exceeds 1. The plots also confirm that at q ≈ 1 the Tsallis‑based entropy aligns closely with previously reported degree‑entropy values, validating the approach.

In the discussion, the authors argue that the q parameter provides a flexible “dial” to explore a spectrum from maximal disorder (high complexity) to maximal order (low complexity) within the same network, thereby uncovering which subsets of nodes (peripheral vs. hub) are most influential for the network’s overall complexity. This capability is absent in traditional entropy measures, which implicitly assume q = 1.

The conclusion emphasizes that Tsallis entropy offers a generalized, tunable framework for assessing network complexity, bridging structural topology and node‑level relational importance. By adjusting q, researchers can tailor the metric to specific analytical goals—such as highlighting vulnerability of peripheral nodes, assessing robustness of hub structures, or tracking phase‑like transitions in evolving networks. The paper suggests future extensions, including optimizing q based on empirical criteria, integrating additional node attributes (e.g., clustering coefficient, betweenness), and applying the method to dynamic or multilayer networks. Overall, the work contributes a theoretically grounded, empirically validated tool that enriches the quantitative toolbox for complex‑network science.